Conditional Probability

Date: 07/18/98 at 22:44:54
From: Carole Black
Subject: Conditional Probability
I will be a beginning math teacher in the fall and will be teaching
Statistics. I am "boning up" on conditional probabilities and I have
a question about an example in the Basic Probability information from
the Ask Dr. Math faq. The example is discussing the independent
events of drawing red or blue marbles. There are 6 blue marbles and 4
red marbles. The discussion goes on to talk about two events, the
second outcome dependent upon the first. The actual example is: But
suppose we want to know the probability of your drawing a blue marble
and my drawing a red one?
Here are the possibilities that make up the sample space:
a. (you draw a blue marble and then I draw a blue marble)
b. (you draw a blue marble and then I draw a red marble)
c. (you draw a red marble and then I draw a blue marble)
d. (you draw a red marble and then I draw a red marble)
The calculation for b is given as:
your probability of drawing a blue marble (3/5) multiplied
by my probability of drawing a red marble (4/9):
3/5 x 4/9 = 12/45 or, reduced, 4/15.
My question is: is this the same thing as P(Red|Blue)? I believe
these are two different things, but I am confused as to how to explain
the difference. For P(Red | Blue) I calculate this probability as:
(4/15)/(6/10) = 4/9.
Can you help clear up my confusion so I can explain this clearly to my
students in the fall?
Thank you,
Carole Black

Date: 07/19/98 at 08:07:58
From: Doctor Anthony
Subject: Re: Conditional Probability
Your second answer P(Red|Blue) = 4/9 is correct
This means that the probability of drawing a Red given that the first
draw was a blue is 4/9. Note the word 'given'. We know before making
the second draw that the first draw was a blue. This must be contrasted
with the probability of red-blue before we start making any draw. The
probability of red-blue is 6/10 x 4/9 = 4/15 and this probability is
calculated before the result of the first draw is known.
The word 'conditional' alerts us to the fact that we are calculating
probabilities 'conditional' on knowing further information partway
through the experiment. These probabilities are also referred to as
'Bayesian' probability, named after the probability theorist Thomas
Bayes (1702-61) who gave this theorem:
P(A and E)
P(E|A) = ----------
P(A)
In other words, if we know that A has occurred, then the sample space
is reduced to the probability of event A, and the denominator for
P(A and E) is not 1 but P(A).
- Doctor Anthony, The Math Forum
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